Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(7,11)(10,12)(15,19)(18,20)(21,22,26,27)(23,30,28,24)(25,33,35,29)(31,32,36,34);(6,9)(10,12)(13,14,17,16)(15,18,19,20)(22,27)(23,28)(29,33)(31,36);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,20)(17,19)(21,23,26,28)(22,24,27,30)(25,31,35,36)(29,32,33,34);(1,2)(3,4)(13,16,17,14)(15,20,19,18)(21,24)(22,28)(23,27)(25,34)(26,30)(29,31)(32,35)(33,36);(5,7)(6,10)(8,11)(9,12)(21,25,26,35)(22,29,27,33)(23,31,28,36)(24,32,30,34)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=f,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4325295);